Nearly K\"ahler submanifolds of a space form

TL;DR

This paper investigates nearly K"ahler submanifolds in space forms, revealing a totally umbilic foliation structure and proving the non-existence of non-homogeneous 6-dimensional nearly K"ahler submanifolds, advancing classification efforts.

Contribution

It establishes a foliation structure on nearly K"ahler submanifolds and proves non-existence results for certain non-homogeneous cases, contributing to their classification.

Findings

Nearly K"ahler submanifolds have a totally umbilic foliation with 6-dimensional leaves.

No non-homogeneous 6-dimensional nearly K"ahler submanifolds exist in space forms.

Progress towards classifying nearly K"ahler hypersurfaces in space forms.

Abstract

In this article we study isometric immersions of nearly K\"ahler manifolds into a space form (specially Euclidean space) and show that every nearly K\"ahler submanifold of a space form has a totally umbilic foliation whose leafs are 6-dimensional nearly K\"ahler manifolds. Moreover using this foliation we show that there is no non-homogeneous 6-dimensional nearly K\"ahler submanifold of a space form. We prove some results towards a classification of nearly K\"ahler hypersurfaces in standard space forms.